Cohomologia de grupos: da construção via resoluções à interpretação via derivações

Abstract

In this work we approach some essential concepts and results of homological algebra, such as the concept of modules, homomorphisms between modules, free modules, modules over the RG ring, exact and semi-exact sequences (we are dealing here with chains complexes, since the cochains complexes have similar results), homomorphism modules and projective modules. In the study of chain complexes, we also approach chain transformations and chain homotopy, essential to show the uniqueness of free and projective resolutions. With these initial concepts of homological algebra, we had the objective of building free and projective resolutions and showing their independence through a homotopy of chains. We also brought, as examples, the normal resolution, the bar resolution and the normalized bar resolution. Given a resolution, which is a chains complex, applying HomR(−,A) we get a cochains complex and thus we are able to construct the cohomology groups of a group G, namely Hn(G,A), with n ∈ Z, which does not depend on the choice of resolution since the functor of the category of modules over RG in the category of abelian groups preserves homotopy of chains. Next, we make a relation of the n-th cohomology groups of G with their groups of (co)invariants and we make some calculations relating Hn(G,A) with AG and with AG. Finally, we perform the interpretation of H1(G,A) via derivation groups and main derivations, and we calculate some special cases, for example, when G = ⟨t⟩ ≈ Z is the infinite cyclic group and when G = ⟨t⟩ ≈ Zn is the finite cyclic group of order n, with n ∈ Z+.